By considering the dimensions of the range or null space, determine the rank and the nullity of the following linear maps:
a) D:Pn --> Pn-1, where D(x^k)=Kx^k-1
b) L:M(2,3) --> M(2,3) where L([a b c; d e f])=[d e f; 0 0 0]
c) Tr:M(3,3) --> R, where Tr(A)=a11+a22+a33 (the trace of A)
The Attempt at a Solution
a)If we rewrite Pn-1 as a column vector and row-reduce, we are left with just 1 in the first row and all the other entries are zeroes. Therefore the nullity is 1 since we have one parameter in the solution space. The rank is 1 - nullity = 0
b)Solving for the nullspace we see that there are 2 parameters in the solution space therefore the nullity is 2 and the rank is 1
c)We have one equation in 3 unknown numbers .The kernel consists of all a11, a22, a33 such that a11+a22+a33 = 0. We can solve that for one of the variables in terms of the other 2. Therefore the nullity is 2 and the rank is 1.
Can you give me a matrix which represents the linear mapping Tr? :S
I am not sure if I did these correctly. I would appreciate any feedback possible. thank you!